A person opens a door by applying a 11-N force perpendicular to it at a distance 0.90m from the hinges. The door is pushed wide open (to 120 ∘ ) in 2.2s How much work was done? What was the average power delivered?
To find the work done in opening the door, we need to calculate the torque exerted on the door and then multiply it by the angle through which the door is rotated.
1. First, let's calculate the torque applied to the door. Torque (τ) is given by the formula:
τ = F * r * sin(θ)
where F is the applied force, r is the distance from the hinge, and θ is the angle between the force and the line connecting the hinge to the point of application.
Given:
Applied force, F = 11 N
Distance from the hinge, r = 0.90 m
Angle, θ = 90° (since the force is perpendicular to the door)
Plugging in these values, we have:
τ = 11 N * 0.90 m * sin(90°)
τ = 11 N * 0.90 m * 1
τ = 9.9 N⋅m
2. Next, we need to calculate the work done by multiplying the torque by the angle of rotation (in radians).
Since the door is pushed wide open to 120°, we convert this angle to radians by multiplying by π/180:
θ_rad = 120° * π/180
θ_rad = 2.09 rad (rounded to two decimal places)
Now we can calculate the work using the formula:
Work (W) = τ * θ_rad
W = 9.9 N⋅m * 2.09 rad
W ≈ 20.67 J (rounded to two decimal places)
So, the work done in opening the door is approximately 20.67 Joules.
To find the average power delivered, we need to divide the total work done by the time taken to do it.
Given:
Time taken, t = 2.2 s
Average power (P) is calculated using the formula:
P = W / t
P = 20.67 J / 2.2 s
P ≈ 9.4 W (rounded to one decimal place)
Therefore, the average power delivered is approximately 9.4 Watts.